Shimano do 2 different bleed kits. One however is just the funnel. The other is significantly more expensive and is a funnel and a syringe.
I got a kit from epic that I use most days in my workshop. Excellent value. Can't recommend them enough.

Shimano do one with a syringe and a wee widget that holds the hose securely on the bleed nipple and the funnel. Think it comes in it's own wee case (mine did anyway).
RRP is £22.99, so not too much really?

Bleeding from the top down is ok. The main problem with top down gravity bleeding is that bubble want to travel upwards. So you may end up using more fluid than you would doing it from the calliper upwards. Make sure you flick the lever a few times to dislodge any bubbles in the lever.

Obviously Shimano say to bleed both ways. After bleeding both ways, another way to help get the last tiny bubble out is when flicking the lever as suggested above also angle the lever body forward and backwards.

Obviously Shimano say to bleed both ways. After bleeding both ways, another way to help get the last tiny bubble out is when flicking the lever as suggested above, also angle the lever body forward and backwards.